From [1]}, the first-order optimality condition of \({\cal L}\)  w.r.t. \(f_k\)  yields
\(f_k(\lambda )= \min \left\lbrace {\sqrt{P_k}}, \frac{w_k \nu _k \sqrt{\phi _k^{(\text{d})}}}{w_k \nu _k^2 \phi _k^{(\text{d})} + \sum \limits _{i=1}^K w_i \nu _i^2 \psi _{ki}^{(\text{d})}+ \lambda \varphi _k^{(\text{d})} }\right\rbrace .\) 
